Calculus Examples

$y = \sin (5 x) + \cos (2 x)$

Step 1

Set $y$ as a function of $x$ .

$f (x) = \sin (5 x) + \cos (2 x)$

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of $\sin (5 x) + \cos (2 x)$ with respect to $x$ is $\frac{d}{d x} [\sin (5 x)] + \frac{d}{d x} [\cos (2 x)]$ .

$\frac{d}{d x} [\sin (5 x)] + \frac{d}{d x} [\cos (2 x)]$

Step 2.2

Evaluate $\frac{d}{d x} [\sin (5 x)]$ .

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Step 2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 2.2.1.1

To apply the Chain Rule, set $u_{1}$ as $5 x$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [5 x] + \frac{d}{d x} [\cos (2 x)]$

Step 2.2.1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [5 x] + \frac{d}{d x} [\cos (2 x)]$

Step 2.2.1.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$\cos (5 x) \frac{d}{d x} [5 x] + \frac{d}{d x} [\cos (2 x)]$

Step 2.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\cos (5 x) (5 \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\cos (5 x) (5 \cdot 1) + \frac{d}{d x} [\cos (2 x)]$

Step 2.2.4

Multiply $5$ by $1$ .

$\cos (5 x) \cdot 5 + \frac{d}{d x} [\cos (2 x)]$

Step 2.2.5

Move $5$ to the left of $\cos (5 x)$ .

$5 \cos (5 x) + \frac{d}{d x} [\cos (2 x)]$

Step 2.3

Evaluate $\frac{d}{d x} [\cos (2 x)]$ .

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Step 2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 2.3.1.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

$5 \cos (5 x) + \frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [2 x]$

Step 2.3.1.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$5 \cos (5 x) - \sin (u_{2}) \frac{d}{d x} [2 x]$

Step 2.3.1.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$5 \cos (5 x) - \sin (2 x) \frac{d}{d x} [2 x]$

Step 2.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$5 \cos (5 x) - \sin (2 x) (2 \frac{d}{d x} [x])$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$5 \cos (5 x) - \sin (2 x) (2 \cdot 1)$

Step 2.3.4

Multiply $2$ by $1$ .

$5 \cos (5 x) - \sin (2 x) \cdot 2$

Step 2.3.5

Multiply $2$ by $- 1$ .

$5 \cos (5 x) - 2 \sin (2 x)$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.27239707, - 0.36860767, - 0.86117382, 1.01576505, - 1.57079632, 1.57079632, 2.12582759$

Step 4

Solve the original function $f (x) = \sin (5 x) + \cos (2 x)$ at $x = 0.27239707$ .

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Step 4.1

Replace the variable $x$ with $0.27239707$ in the expression.

$f (0.27239707) = \sin (5 (0.27239707)) + \cos (2 (0.27239707))$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $5$ by $0.27239707$ .

$f (0.27239707) = \sin (1.36198538) + \cos (2 (0.27239707))$

Step 4.2.1.2

Multiply $2$ by $0.27239707$ .

$f (0.27239707) = \sin (1.36198538) + \cos (0.54479415)$

Step 4.2.2

The final answer is $\sin (1.36198538) + \cos (0.54479415)$ .

$\sin (1.36198538) + \cos (0.54479415)$

Step 5

Solve the original function $f (x) = \sin (5 x) + \cos (2 x)$ at $x = - 0.36860767$ .

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Step 5.1

Replace the variable $x$ with $- 0.36860767$ in the expression.

$f (- 0.36860767) = \sin (5 (- 0.36860767)) + \cos (2 (- 0.36860767))$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $5$ by $- 0.36860767$ .

$f (- 0.36860767) = \sin (- 1.84303835) + \cos (2 (- 0.36860767))$

Step 5.2.1.2

Multiply $2$ by $- 0.36860767$ .

$f (- 0.36860767) = \sin (- 1.84303835) + \cos (- 0.73721534)$

Step 5.2.2

The final answer is $\sin (- 1.84303835) + \cos (- 0.73721534)$ .

$\sin (- 1.84303835) + \cos (- 0.73721534)$

Step 6

Solve the original function $f (x) = \sin (5 x) + \cos (2 x)$ at $x = - 0.86117382$ .

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Step 6.1

Replace the variable $x$ with $- 0.86117382$ in the expression.

$f (- 0.86117382) = \sin (5 (- 0.86117382)) + \cos (2 (- 0.86117382))$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $5$ by $- 0.86117382$ .

$f (- 0.86117382) = \sin (- 4.3058691) + \cos (2 (- 0.86117382))$

Step 6.2.1.2

Multiply $2$ by $- 0.86117382$ .

$f (- 0.86117382) = \sin (- 4.3058691) + \cos (- 1.72234764)$

Step 6.2.2

The final answer is $\sin (- 4.3058691) + \cos (- 1.72234764)$ .

$\sin (- 4.3058691) + \cos (- 1.72234764)$

Step 7

Solve the original function $f (x) = \sin (5 x) + \cos (2 x)$ at $x = 1.01576505$ .

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Step 7.1

Replace the variable $x$ with $1.01576505$ in the expression.

$f (1.01576505) = \sin (5 (1.01576505)) + \cos (2 (1.01576505))$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $5$ by $1.01576505$ .

$f (1.01576505) = \sin (5.07882527) + \cos (2 (1.01576505))$

Step 7.2.1.2

Multiply $2$ by $1.01576505$ .

$f (1.01576505) = \sin (5.07882527) + \cos (2.0315301)$

Step 7.2.2

The final answer is $\sin (5.07882527) + \cos (2.0315301)$ .

$\sin (5.07882527) + \cos (2.0315301)$

Step 8

Solve the original function $f (x) = \sin (5 x) + \cos (2 x)$ at $x = - 1.57079632$ .

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Step 8.1

Replace the variable $x$ with $- 1.57079632$ in the expression.

$f (- 1.57079632) = \sin (5 (- 1.57079632)) + \cos (2 (- 1.57079632))$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $5$ by $- 1.57079632$ .

$f (- 1.57079632) = \sin (- 7.85398163) + \cos (2 (- 1.57079632))$

Step 8.2.1.2

Multiply $2$ by $- 1.57079632$ .

$f (- 1.57079632) = \sin (- 7.85398163) + \cos (- 3.14159265)$

Step 8.2.2

The final answer is $\sin (- 7.85398163) + \cos (- 3.14159265)$ .

$\sin (- 7.85398163) + \cos (- 3.14159265)$

Step 9

Solve the original function $f (x) = \sin (5 x) + \cos (2 x)$ at $x = 1.57079632$ .

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Step 9.1

Replace the variable $x$ with $1.57079632$ in the expression.

$f (1.57079632) = \sin (5 (1.57079632)) + \cos (2 (1.57079632))$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Multiply $5$ by $1.57079632$ .

$f (1.57079632) = \sin (7.85398163) + \cos (2 (1.57079632))$

Step 9.2.1.2

Multiply $2$ by $1.57079632$ .

$f (1.57079632) = \sin (7.85398163) + \cos (3.14159265)$

Step 9.2.2

The final answer is $\sin (7.85398163) + \cos (3.14159265)$ .

$\sin (7.85398163) + \cos (3.14159265)$

Step 10

Solve the original function $f (x) = \sin (5 x) + \cos (2 x)$ at $x = 2.12582759$ .

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Step 10.1

Replace the variable $x$ with $2.12582759$ in the expression.

$f (2.12582759) = \sin (5 (2.12582759)) + \cos (2 (2.12582759))$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Multiply $5$ by $2.12582759$ .

$f (2.12582759) = \sin (10.62913799) + \cos (2 (2.12582759))$

Step 10.2.1.2

Multiply $2$ by $2.12582759$ .

$f (2.12582759) = \sin (10.62913799) + \cos (4.25165519)$

Step 10.2.2

The final answer is $\sin (10.62913799) + \cos (4.25165519)$ .

$\sin (10.62913799) + \cos (4.25165519)$

Step 11

The horizontal tangent lines on function $f (x) = \sin (5 x) + \cos (2 x)$ are $y = \sin (1.36198538) + \cos (0.54479415), y = \sin (- 1.84303835) + \cos (- 0.73721534), y = \sin (- 4.3058691) + \cos (- 1.72234764), y = \sin (5.07882527) + \cos (2.0315301), y = \sin (- 7.85398163) + \cos (- 3.14159265), y = \sin (7.85398163) + \cos (3.14159265), y = \sin (10.62913799) + \cos (4.25165519)$ .

$y = \sin (1.36198538) + \cos (0.54479415), y = \sin (- 1.84303835) + \cos (- 0.73721534), y = \sin (- 4.3058691) + \cos (- 1.72234764), y = \sin (5.07882527) + \cos (2.0315301), y = \sin (- 7.85398163) + \cos (- 3.14159265), y = \sin (7.85398163) + \cos (3.14159265), y = \sin (10.62913799) + \cos (4.25165519)$

Step 12